Constant curvature

In mathematics, constant curvature is a concept from differential geometry. Here, curvature refers to the sectional curvature of a space (more precisely a manifold) and is a single number determining its local geometry.^[1] The sectional curvature is said to be constant if it has the same value at every point and for every two-dimensional tangent plane at that point. For example, a sphere is a surface of constant positive curvature.

Classification

The Riemannian manifolds of constant curvature can be classified into the following three cases:

• Elliptic geometry – constant positive sectional curvature
• Euclidean geometry – constant vanishing sectional curvature
• Hyperbolic geometry – constant negative sectional curvature.

Properties

• Every space of constant curvature is locally symmetric, i.e. its curvature tensor is parallel ${\displaystyle \nabla \mathrm {R} =0}$.
• Every space of constant curvature is locally maximally symmetric, i.e. it has ${\displaystyle {\frac {1}{2}}n(n+1)}$ number of local isometries, where ${\displaystyle n}$ is its dimension.
• Conversely, there exists a similar but stronger statement: every maximally symmetric space, i.e. a space which has ${\displaystyle {\frac {1}{2}}n(n+1)}$ (global) isometries, has constant curvature.
• (Killing–Hopf theorem) The universal cover of a manifold of constant sectional curvature is one of the model spaces:
  • sphere (sectional curvature positive)
  • plane (sectional curvature zero)
  • hyperbolic manifold (sectional curvature negative)
• A space of constant curvature which is geodesically complete is called space form and the study of space forms is intimately related to generalized crystallography (see the article on space form for more details).
• Two space forms are isomorphic if and only if they have the same dimension, their metrics possess the same signature and their sectional curvatures are equal.

References

Further reading

• Moritz Epple (2003) From Quaternions to Cosmology: Spaces of Constant Curvature ca. 1873 — 1925, invited address to International Congress of Mathematicians
• Frederick S. Woods (1901). "Space of constant curvature". The Annals of Mathematics. 3 (1/4): 71–112. doi:10.2307/1967636. JSTOR 1967636.